The long term goal of this research is to explain the mechanism of hypertonic urine formation in the renal medulla of mammals. The proposed work consists in the derivation and execution of a large model of medullary structure and function. The model is cast as a non-linear boundary value problem. The difficulty with existing models is that they fail to achieve reasonable predictions of urinary osmolality without unrealistic simplifications in model structure which are made to achieve greater efficiency and accuracy in the methods of solution. The state of affairs means that hypotheses of medullary function still lack the validation they need. Recent developments in numerical methods can now be applied to more accurate procedures for solving boundary value problems, and one of the goals of this research will be to apply these methods to more realistic models. Recent results in anatomical research suggest a regular arrangement of blood vessels and tubules in 3 dimensions, whereas existing countercurrent models have a single spatial dimension. The functional significance of this regular arrangement will be assessed by deriving and solving a model that captures the 3-dimensional arrangement. This larger, more detailed model, will be the major work of this project.